TREE sequence
The TREE sequence is a very fast-growing function arising out of , devised by Harvey Friedman.Adam Goucher, TREE(3) and impartial gamesFriedman's post on FOM mailing list Definition Suppose we have a sequence of k''- T1, T2 ... with the following properties: # Each tree T''i has at most i'' vertices. # No tree is into any tree following it in the sequence. states that such a sequence cannot be infinite. Harvey Friedman expanded on this by asking the question: given ''k, what is the maximum length of such a sequence? This maximal length is a function of k'', dubbed TREE(''k). The first two values are TREE(1) = 1 and TREE(2) = 3. The next value, TREE(3), is so large that it surpasses many large combinatorial constants like n(4); it is the subject of extensive research regarding its size. Since TREE(3) > n(4), a very weak lower bound for it is AA(187196)(1) using the Ackermann function. It has also been provenHow large is TREE(3) that TREE(3) > (5) for any "reasonable amount" X. It can be shown that TREE(n) grows faster than \(f_{\vartheta(\Omega^\omega)}(n)\) in the fast-growing hierarchy. The current limit of BEAF is believed to be around \(f_{\varphi(2,0,0)}(n)\), so TREE(3) probably surpasses many or all of Bowers' numbers, including meameamealokkapoowa oompa. It is certainly much smaller than Rayo's number, however — TREE(n) is still a computable function. Chris Bird has shown that \(\text{TREE}(3) > \lbrace3, 6, 3 [1 \neg 1,2 2] 2\rbrace\), using his array notation.Chris Bird, Beyond Bird's Nested Arrays II, page 10 Explanation Trees are tricky to visualize without drawing them out, so we shall devise a simpler way of representing them. Consider a language which has various kinds of parentheses such as (), [], {} etc. Parentheses always come in pairs and can nest within each other. Within a larger node, they may be concatenated. For example, the following strings are valid in this language: [] ([]) {()()} [(){[[]]}(){(())[]}] A string A'' is a '''substring' of another string B'' if ''B is of the form "XAY," where X and Y are arbitrary and possibly empty strings. Informally, this means that A can be found inside B with no modifications. For example, {[[]]} and (()) are substrings of the fourth string above. {} is not a substring, since it doesn't appear without modifications. With all this in mind, we can create an equivalent definition of TREE(k''). Suppose we have a sequence of strings with the following properties: # You may only use ''k types of brackets. # The first string has at most one pair of brackets, the second string has at most two pairs of brackets, the third string has at most three pairs of brackets, etc. # No string is a substring of a later string. TREE(k'') is the maximal length of the sequence. For ''k = 1, the optimal sequence has only one member: (). For k'' = 2, the optimal sequence has only two members: (), then ([]), then []. Weak tree function Weak tree function, tree(n) is defined to be longest sequence of 1-labelled trees such that: # Every tree at position ''k (for all k'') has no more that ''k+''n'' vertices. # No tree is homeomorphically embeddable into any tree following it in the sequence. Adam P. Goucher has shown following properties of this function: # tree(n) has growing level \(\vartheta(\Omega^\omega)\) ( ) in the fast-growing hierarchy. # TREE(3) > treetreetreetreetree8(7)(7)(7)(7)(7) Also found larger lower bound for TREE(3). Define \(tree_2(n)=tree^n(n)\) and \(tree_3(n)=tree_2^n(n)\). Then TREE(3) > tree\(_3\) (tree\(_2\) (tree(8))), where \(tree_3 (tree_2 (tree(8) ) ) \) is about treetreetree...tree(n)...(n)(n)(n) with n exponents, where \(n = tree_2 (tree(8)) = tree^{tree(8)} (tree(8)) \).How does TREE(3) grow to get so big need laymen explanation Sources See also Category:Numbers Category:Functions Category:Theorems